In three dimensions, a unit vector will be distinguished by 2 angles. It could be a polar angle ##\theta## and an azimuthal angle ##\phi## or some other two numbers to define a direction (sometimes the designation is flipped). In spherical polar coordinates a unit vector will be ##\hat{u}=(\sin\phi\cos\theta,\sin\phi\cos\phi,\sin\phi)##

I don't understand what it means: "a vector making an angle of −π/4 with ∇f(3,1,2)."? In 3-D space, there are an infinity of vectors making a specific angle with respect to another vector...they form a cone.

I don't understand what it means: "a vector making an angle of −π/4 with ∇f(3,1,2)."? In 3-D space, there are an infinity of vectors making a specific angle with respect to another vector...they form a cone.

You do not have to think about angles to understand the idea of a directional derivative. Let x and v be in Rn, v be a unit vector and t in R . The directional derivative of f at a in the direction v is just the derivative of the single variable function h(t) = f(a + tv) at t=0 (that is h'(0) ).

This gives the definition of the directional derivative without discussing the gradient. And in fact some directional derivatives can exist without f having a defined gradient. Say f is differentible with respect to x but not with respect to y. However if f is differentible then the direction derivative of f at a in the direction v is grad(f)(a)*v.

Given function f(x, y, z), with gradient [itex]\nabla f[/itex], we can talk about the "directional derivative" in the direction of unit vector [itex]\vec{v}[/itex] as the dot product: [itex]\nabla f\cdot \vec{v}[/itex].

In three dimensions, a direction cannot be specified by a single angle. We need two angles such as the "[itex]\theta[/itex]" and "[itex]\phi[/itex]" used in spherical coordinates. Or we can use the "direction cosines", the cosines of the angles the direction makes with the three coordinate axes: [itex]\theta_x[/itex] is the angle a line in the given direction makes with the x-axis, [itex]\theta_y[/itex] the angle it makes with the y-axis, and [itex]\theta_z[/itex], the angle it makes with the z-axis. Of course, those three angles are not idependent. We can show that we must have [itex]cos^2(\theta_x)+ cos^2(\theta_y)+ cos^2(\theta_z)= 1[/itex] which means that the vector [itex] cos(\theta_x)\vec{i}+ cos(\theta_y)\vec{j}+ cos(\theta_z)\vec{k}[/itex] is the unit vector in that direction.

That is, the "directional derivative" of f(x, y, z) in the direction that makes angles [itex]\theta_x[/itex], [itex]\theta_y[/itex], and [itex]\theta_z[/itex] with the x, y, and z axes, respectively, is given by [itex]\nabla f\cdot (cos(\theta_x)\vec{i}+ cos(\theta_y)\vec{j}+ cos(\theta_z)\vec{k})= f_x cos(\theta_x)+ f_y cos(\theta_y)+ f_z cos(\theta_z)[/itex].